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- W66478277 abstract "We generalize an old problem and its partial solution of Pólya (Pólya in Elem. Math. 13, 40–41 ()) to the n-dimensional setting. Given a plane domain Ω⊂ℝ2, Pólya asked in 1958 for the shortest bisector of Ω, that is for the shortest line segment l(Ω) which divides Ω into two subsets of equal area. He claimed that among all centrosymmetric domains of given area l(Ω) becomes longest for a disk. His proof, however, does not seem to be valid for domains that are not starshaped with respect to the center of Ω. In the present note we provide two proofs that it suffices to restrict attention to starshaped sets. Moreover we state and prove a related inequality in ℝn. Given the volume of a measurable set Ω with finite Lebesgue measure, only a ball centered at zero maximizes the length of the shortest line segments running through the origin. In this sense the ball has the longest shortest piercing." @default.
- W66478277 created "2016-06-24" @default.
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- W66478277 date "2012-01-01" @default.
- W66478277 modified "2023-10-09" @default.
- W66478277 title "The Longest Shortest Piercing" @default.
- W66478277 cites W1535130726 @default.
- W66478277 doi "https://doi.org/10.1007/978-3-0348-0249-9_8" @default.
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